Possibility of long-range order in clean mesoscopic cylinders

A microscopic Hamiltonian of the magnetostatic interaction is discussed. This long-range interaction can play an important role in mesoscopic systems leading to an ordered ground state. The self-consistent mean field approximation of the magnetostatic interaction is performed to give an effective Hamiltonian from which the spontaneous, self-sustaining currents can be obtained. To go beyond the mean field approximation the mean square fluctuation of the total momentum is calculated and its influence on self-sustaining currents in mesoscopic cylinders with quasi-1D and quasi-2D conduction is considered. Then, by the use of the microscopic Hamiltonian of the magnetostatic interaction for a set of stacked rings, the problem of long-range order is discussed. The temperature $T^{*}$ below which the system is in an ordered state is determined.